A note on the structure of $\frac{\mathbb{F}_qS_5}{J(\mathbb{F}_qS_5)}$

Year 2026, Issue: Advanced Online Publication, 1 - 8

Pramod Kanwar , Yogesh Kumar Rajendra Kumar Sharma

https://doi.org/10.24330/ieja.1823884

Abstract

A complete algebraic structure of $\frac{\mathbb{F}_qS_5}{J(\mathbb{F}_qS_5)}$, where $\mathbb F_q=\mathbb F_{p^n}$ is a finite field with characteristic $p$ (a prime) and $J(\mathbb{F}_qS_5)$ is the Jacobson radical of the group algebra $\mathbb{F}_qS_5$ is given.

Keywords

Group algebra , Wedderburn decomposition , unit group

References

• A. A. Bovdi and A. Szakacs, The unitary subgroup of the multiplicative group of the modular group algebra of a finite abelian $p$-group, Math. Zametki, 45(6) (1989), 23-29.
• L. Creedon and J. Gildea, Unitary units of the group algebra $\mathbb{F}_{2^k}Q_8$, Internat. J. Algebra Comput., 19(2) (2009), 283-286.
• C. W. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Pure and Applied Mathematics, $XI$, Interscience Publishers (a division of John Wiley \& Sons, Inc.), New York-London, 1962.
• A. Dholakia, Introduction to Convolutional Codes with Applications, Kluwer Academic Publishers, Boston-London, 1994.
• R. A. Ferraz, Simple components of center of $FG/J(FG)$, Comm. Algebra, 36(9) (2008), 3191-3199.
• J. Gildea, The structure of the unit group of the group algebra $\mathbb{F}_{2^{k}}A_{4}$, Czechoslovak. Math. J., 61(2) (2011), 531-539.
• J. Gildea and F. Monaghan, Units of some group algebras of groups of order 12 over any finite field of characteristic 3, Algebra Discrete Math., 11(1) (2011), 46-58.
• T. Hurley, Group rings and ring of matrices, Int. J. Pure Appl. Math., 31(3) (2006), 319-335.
• T. Hurley, Convolutional codes from units in matrix and group rings, Int. J. Pure Appl. Math., 50(3) (2009), 431-463.
• G. D. James, The Representation Theory of the Symmetric Groups, Lecture Notes in Mathematics, 682, Springer, Berlin, 1978.
• G. Karpilovsky, The Jacobson Radical of Group Algebras, North-Holland Mathematics Studies, 135, North-Holland Publishing Co., Amsterdam, 1987.
• M. Khan, R. K. Sharma and J. B. Srivastava, The unit group of $\mathbb{F}S_4$, Acta Math. Hungar., 118 (2008), 105-113.
• Y. Kumar, R. K. Sharma and J. B. Srivastava, The structure of the unit group of the group algebra $\mathbb{F}S_5$ where $\mathbb{F}$ is a finite field with char$(\mathbb{F})=p>5$, Acta Math. Acad. Paedagog. Nyhazi, 33(2) (2017), 187-191.
• S. Maheshwari and R. K. Sharma, The unit group of group algebra $\mathbb{F}_qSL(2,\mathbb Z_3)$, J. Algebra Comb. Discrete Struct. Appl., 3(1) (2016), 1-6.
• N. Makhijani, R. K. Sharma and J. B. Srivastava, A note on units in $\mathbb{F}_{p^m}D_{2p^n}$, Acta Math. Acad. Paedagog. Nyhazi, 30(1) (2014), 17-25.
• N. Makhijani, R. K. Sharma and J. B. Srivastava, The unit group of algebra of circulant matrices, Int. J. Group Theory, 3(4) (2014), 13-16.
• N. Makhijani, R. K. Sharma and J. B. Srivastava, The unit group of finite group algebra of a generalized dihedral group, Asian-Eur. J. Math., 7(2) (2014), 1450034 (5 pp).
• N. Makhijani, R. K. Sharma and J. B. Srivastava, Units in $\mathbb{F}_{2^{k}}D_{2n}$, Int. J. Group Theory, 3(3) (2014), 25-34.
• N. Makhijani, R. K. Sharma and J. B. Srivastava, A note on the structure of $\mathbb{F}_{p^{k}}A_{5}/J(\mathbb{F}_{p^{k}}A_{5})$, Acta Sci. Math. (Szeged), 82 (2016), 29-43.
• C. Polcino Milies and S. K. Sehgal, An Introduction to Group Rings, Algebra and Applications, 1, Kluwer Academic Publishers, Dordrecht, 2002.
• R. K. Sharma, J. B. Srivastava and M. Khan, The unit group of $\mathbb{F}S_3$, Acta Math. Acad. Paedagog. Nyhazi, 23(2) (2007), 129-142.
• R. K. Sharma, J. B. Srivastava and M. Khan, The unit group of $\mathbb{F}A_4$, Publ. Math. Debrecen, 71 (2007), 21-26.
• R. K. Sharma and P. Yadav, The unit group of $\mathbb{Z}_p Q_8$, Algebras Groups Geom., 25(4) (2008), 425-429.
• R. K. Sharma and P. Yadav, Unit group of algebra of circulant matrices, Int. J. Group Theory, 2(4) (2013), 1-6.